Another Red Hatter here to give a recommendation for Red Hat. All of my work is in the open. Furthermore I can say just about all code written at RH is upstream-first. Red Hat has a great culture and is very remote friendly. (My team is fairly distributed)

Is it possible to work at Red Hat without using any proprietary software, let alone write any?

It's not possible to work ANYWHERE without using proprietary software, running on patended hardware, coming partially from China.

Hardware and patents are both unrelated to my question.

Not only possible, I'm pretty sure it's impossible to work exclusively on closed-source at Red Hat.

I would say it depends on the specific job. :)

Happy to talk about it, I am just a bit afraid that it will flood the site and make it hard to see other offers.

I think the majority of the jobs in our technology category would qualify:

https://www.redhat.com/en/jobs/categories/technology

Again, happy to coordinate a direct contact.

You can get me at jwildebo [at] redhat dot com